grid dispersion and filtering in broadband acoustic

Grid dispersion and filteringin broadband acoustic emission (AE) source modeling

Siddhesh Raorane1 , Paweł Paćko2

1 AGH University of Science and Technology, Academic Centre for Materials and Nano-technology, Krakow

2 AGH University of Science and Technology, Faculty of Mechanical Engineering and Robotics,Department of Robotics and Mechatronics, Krakow

Abstract: The spatial grid of numerical models of acoustic emission (AE) sources acts as a spa-tial filter for elastic wave signals. The filtering effects are particularly prominent for short-time,broadband signals – typical for AE. In this paper, we investigate the filtering influence of spatialdiscretization (meshing) on broadband AE source modeling. The AE source – generating AEspropagating as elastic waves – was modeled using cohesive zone approach, and the numericalsimulations were performed in commercial FEM software COMSOL Multiphysics. Resultswere processed using Fast Fourier Transform, filtered, and subsequently analyzed in terms ofthe filtering effects of spatial discretization on AE source modeling. In this paper, it is shownthat spatial grids in numerical models effectively work like low-pass filters with the cut-offfrequency corresponding to the numerical Brillouin zone. The latter induces short wavelengthlimitation, and the frequency components near (below) the zone edge are amplified in magni-tude. It was found that the amplified frequencies represent numerical errors. Also, it wasinferred that the filtering effect of spatial discretization can mask the AE source characteristicsand affect the quality of the results.Keywords: acoustic emission source, finite element modeling, cohesive zone approach, griddispersion, signal processing, elastic waves

DYSPERSJA SIATKI I FILTRACJA SYGNAŁÓWW MODELOWANIU SZEROKOPASMOWYCH ŹRÓDEŁ EMISJI AKUSTYCZNEJ

Streszczenie: Siatki przestrzenne modeli numerycznych mają własności filtrów, ujawniające sięw szczególności w krótkotrwałych, szerokopasmowych zjawiskach dynamicznych typowych dlaemisji akustycznej (AE). W artykule poddano analizie wpływ filtrowania, spowodowanego siatkąprzestrzenną, na modelowe sygnały AE. Źródło emisji akustycznej zamodelowane zostało z wyko-rzystaniem stref kohezji, a symulacje wykonano w oprogramowaniu COMSOL Multiphysics. Wy-niki uzyskane z dynamicznej analizy przejściowej poddano analizie, wykorzystując szybką trans-formatę Fouriera oraz filtrację w dziedzinie częstotliwości, a następnie zbadano wpływ siatkiprzestrzennej na zawartość częstotliwościową uzyskanych sygnałów. W artykule pokazano, żesiatki mają charakter dolnoprzepustowych filtrów przestrzennych o częstotliwości odcięciarównej brzegowi numerycznej strefy Brillouina dla badanego modelu. Strefa ta uniemożliwiapropagację fal krótkich, których częstotliwość jest wyższa od jej granicy, lecz również wzmacniaamplitudy fal o częstotliwościach niższych. Te ostatnie, zdeformowane komponenty częstotli-wościowe, mają charakter błędów numerycznych i zmieniają obraz fizycznych cech źródła emisjiakustycznej oraz pogarszają jakość uzyskiwanych wyników symulacji.Słowa kluczowe: źródła emisji akustycznej, metoda elementów skończonych, strefy kohezji, dys-persja siatki, przetwarzanie sygnałów, fale sprężyste

https://doi.org/10.7494/978-83-66727-48-9_10

https://orcid.org/0000-0002-1847-7830

https://orcid.org/0000-0002-8962-9969

https://doi.org/10.7494/978-83-66727-48-9_10

150 S. Raorane, P. Paćko

1. Introduction

Elastic waves find applications across various fields of engineering and science(Cheeke 2017). They are frequently used in mining and geological engineering owingto two unique phenomena, namely, reflection and distortion of the signal whenencountering a change in material property (e.g. a void), and emission of elastic waveswhen distributed or localized micro-displacements occur in a material.

Oil and gas explorations have been employing guided elastic waves to determinethe presence of reservoirs (see e.g. “Reflection Seismology”) wherein elastic waves areinduced in the Earth with the help of transducers, and the modified and/or reflectedwaves are picked up by special sensors. Elastic waves also accompany exploration pro-cesses and are, for instance, emitted as acoustic emission (AE) signals during the drill-ing process thus paving the path for the application of AE techniques (Grosse andOhtsu 2008) in drilling/mining industry. The AE based techniques find far more appli-cations than the guided waves’ based methods in the drilling and mining industries, andare popularly used for prediction, estimation, monitoring and diagnosis (Khoshoueiand Bagherpour 2019). A drill bit-rock interface monitoring technology capable ofproviding the operator with live information about the condition of the down-hole drillbit with acoustic emissions was outlined in Karakus and Perez (2014). A robust methodfor drilling monitoring using AE was proposed in Le Moal et al. (2012). Experimentsshowing the robustness and usefulness of the proposed method in drilling monitoringhave been reported. More recently, in Bejger and Piasecki (2020), AEs were used fordiagnosing high pressure mud pumps used on drilling rigs. The method reported inBejger and Piasecki (2020) was verified by experiments conducted on a NOV (NationalOilwell Varco)-made triplex 14-P-220 mud pump (mounted in the drillship).

AE-based techniques require extraction of the characteristics of the recorded AEsignals making signal processing one of their important aspects. In Rao and Subraman-yam (2008) the AE signals were analyzed using the wavelet transform. A combinedmethod of Winger-Ville distribution signal processing method with a theoreticalvelocity dispersion model for AE source location in dispersive media was reportedby Kim et al. (2013). In Yu et al. (2006), a study on the failure detection of compositematerials using AE was reported wherein the AE signals were analyzed using FastFourier Transform (FFT). This short list is, of course, not exhaustive for signal pro-cessing methods for AE signals.

Although the application of AE is very popular in the drilling and mining indus-tries, the dispersive nature of AE signals makes the correlation between the AE sourceand waveform features still unclear. Thus, AE based methods often require thoroughand robust designing, prototyping, and testing, which are iterative and expensive. Nu-

Grid dispersion and filtering in broadband acoustic emission (AE) source modeling 151

merical modeling of AE diminishes the dependence on expensive and time-consumingphysical experiments making it an attractive and popular tool for predicting complexwavefields and help in the development of robust and reliable AE techniques. Model-ing of buried monopole and dipole AE sources with a FEM technique was reportedin Hamstad et al. (1999). An AE source model based on FEM approach wasreported by Sause and Richler (2015). The authors in modeled the AE source asa crack based on cohesive zone element approach (Sause and Richler 2015).

Most of numerical models of AE sources require a spatial numerical grid (mesh).The spatial grid acts as a spatial filter for elastic wave signals. The filtering effectsare particularly prominent for short-time, broadband signals that are typical for AE.In this paper, we investigate the filtering influence of spatial discretization (meshing)on AE source modeling using three different mesh configurations. Commercial FEMsoftware COMSOL Multiphysics was employed to perform numerical simulations.The AE source – generating AEs propagating as elastic waves – was modeled usingcohesive zone approach.

The paper is organized as follows. First, the modeling approach adopted in thispaper is described. Next, results for three mesh configurations are presented, followedby their processing using Fast Fourier Transform (FFT), and filtering. Subsequently,the original and processed results are analyzed and discussed in terms of the filteringeffects of the spatial discretization on AE source modeling. Lastly, conclusions aredrawn from the analyzes.

2. Model description

The COMSOL Multiphysics commercial FEM software was employed to performnumerical simulations. All calculations presented herein were performed usingthe Structural Mechanics (SM) module of COMSOL. The Low-Reflecting Boundary(LRB) feature of the SM module was employed to absorb far-field outgoing waves.The LRB acts as an absorbing boundary layer, similar to a perfectly matched layer,preventing edge/boundary-reflected waves from interfering with source emissionswithin the analysis domain.

All simulations were performed under 2-D plane-strain conditions. The specimengeometry consisted of a notched square aluminum block with edge length of 0.5 m.The acoustic emission source was modeled using the cohesive zone approach. In order toactivate crack propagation, a point load was applied at the tip of the cohesive zone.The geometrical setup consisting of the cohesive zone and the point load can be seenin Figure 1a. The advancing crack propagation induces acoustic emissions propagatingas elastic waves, as shown in Figure 1b.


Fig. 1. The geometrical setup – used in the FEM simulations – consisting of the cohesive zoneand the point load (a). Advancing crack propagation inducing acoustic emissions propagating

as elastic waves for M0 mesh configuration (coarse mesh) at a selected time instant (b).Please note that model distortion was rescaled for illustration purposes

Spatial numerical grid (mesh) of the model acts as a spatial filter for elastic wavessignals. These filtering effects are particularly prominent for short-time, broadband signalstypical for acoustic emission. To investigate the influence of spatial discretization(meshing) on the generated AE signals, the same model geometry was discretized withthree different mesh configurations. The resultant element sizes varied from 0.01 min M0 mesh (coarse mesh), 0.005 m in M1 mesh (fine mesh), to 0.002 m in M2 mesh(finest mesh), as shown in Figure 2. Please note that the subscripts 0, 1 and 2 are cho-sen only to represent the resolution of each mesh configuration, with 0 representingthe lowest and 2 representing the highest resolutions. All the three mesh configura-tions can be seen in Figure 2; the coarse, fine and finest meshes being illustratedin Figures 2a, 2b and 2c, respectively. Please note that in each mesh configuration,the element size was kept constant throughout the model to avoid spurious/artificialreflections typical for meshes with varying spatial discretization (Raorane et al. 2021).

The time integration was carried out with the COMSOL built-in generalized--alpha time stepping scheme with a maximum step constraint of 0.01 μs used for all thenumerical simulations. The maximum step constraint value was chosen small enoughin order to remain well below the Nyquist limit of the propagating AE signals,i.e. in order to represent all relevant temporal frequency components of the generatedsignal – allowing for attributing signal distortions to the spatial grid dispersion only.

a) b)


The total simulation time was 80 μs for all configurations, and AE signals were record-ed at a sensor point located at the same geometrical position for all three mesh config-urations as shown in Figure 2. Please note that the horizontal and vertical velocities,instead of displacements, were recorded at the sensor point to avoid the DC (staticdisplacement) component.

Fig. 2. Illustration of the different mesh configurations used in this paper: a) M0 mesh configu-ration with element size 0.01 m (coarse mesh); b) M1 mesh configuration with element size

0.005 m (fine mesh); c) M2 mesh configuration with element size 0.002 m (finest mesh)

The simulation starts with a gradually-increasing point-load at each notch face,inducing decohesion and leading to the generation of acoustic emissions. The latterelastic waves are recorded at the sensor point. Example AEs from the source for M0

configuration at a selected time instant can be seen in Figure 1b (please note that modeldistortion was rescaled for illustration purposes).

3. Results

The elastic waves, emitted as AE signals, generated by the cohesive zone wererecorded at the sensor point for all the three mesh configurations at all the time stepsand compared. Figure 3 shows all the recorded AE signals. The velocities of the AEsignals were recorded to avoid the DC shift in displacements. It can be clearly observedfrom Figure 3 that as the mesh resolution becomes finer, i.e., as the element size of themesh decreases, the time domain signals smooths. Thus, the resulting signals are themost coarse for M0 – represented by Figure 3a and the smoothest for M2 – representedby Figure 3c. It should be emphasized that the time integration procedure type andparameters were the same, hence signal distortions are only due to spatial discretiza-tion. It can be observed for improving mesh accuracy (decreasing element size) that

a) b) c)


the acquired signals display more high frequency components, as expected. Interest-ingly, however, less moderate-frequency waves are present for finer meshes (see thesubstantial amplitude variations in Figures 3a and 3b). The latter is in contrast withthe anticipated signal shape convergence, where higher frequency components wouldbe added on top of the signal when shorter waves are allowed by smaller elements.

Fig. 3. The AE signals – generated by the cohesive zone – recorded at the sensor pointfor all the three mesh configurations at all the time steps: a) AE signals recorded for M0

mesh configuration (coarse mesh) at all the time steps; b) AE signals recorded for M1 meshconfiguration (fine mesh) at all the time steps; c) AE signals recorded for M2 mesh configuration

(finest mesh) at all the time steps. It can be observed that as the mesh resolution becomesfiner, i.e., as the element size of the mesh decreases, the time domain signals smooths

3.1. Fast Fourier Transform (FFT)

In order to investigate the results in the frequency domain, Fast Fourier Trans-form (FFT) of the results presented in Figure 3 was performed. It must be noted thatthe FFT of the horizontal and vertical velocities give very similar results for the sensor

a) b)

c)


point location chosen here, and thus the FFT of either one of the velocities can beperformed in order to investigate the influence of the grid. In this paper, only the hor-izontal velocities were analyzed, as seen in Figure 4. It can be clearly observed fromFigure 4 that as the mesh resolution becomes finer, the grid cut-off frequency increasessuch that for M0 (coarse mesh) the cut-off frequency is approximately 0.9 kHz, for M1

(fine mesh) the grid cut-off frequency is approximately 1.8 kHz, and for M2 (finestmesh) the grid cut-off frequency is approximately 4.4 kHz, which is very near thetime-step constrained Nyquist limit of 4.5 kHz. It can also be observed from Figure 4that at very low frequencies (<0.2 kHz), the FFT results for all the three configurationsare the same.

Fig. 4. Fast Fourier Transform (FFT) of the horizontal velocities of all the three mesh configurations presented in Figure 3: a) FFT of the horizontal velocities of M0 mesh

configuration (coarse mesh) presented in Figure 3a; b) FFT of the horizontal velocities of M1mesh configuration (fine mesh) presented in Figure 3b; c) FFT of the horizontal velocitiesof M2 mesh configuration (finest mesh) presented in Figure 3c. It can be clearly observed

that as the mesh resolution becomes finer, the grid cut-off frequency increases

a) b)

c)


3.2. Filtering

To standardize our study with respect to frequency, the FFT results obtainedin Figure 4 were passed through a low-pass filter, which was based on the cut-off fre-quency of the coarse mesh model M0 represented by Figure 4a. Figure 5 showsthe filtered FFT of all the mesh configurations, and it can be clearly seen that thecut-off frequency of all the configurations is now the same. The magnitude of the fre-quencies, however, is different for different configurations. The different magnitudesof frequencies can be clearly observed in Figure 5b, which represents a close-up view ofFigure 5a.

Fig. 5. Filtered FFT of all the three mesh configurations (a). A close-up view of Figure5a (b).It can be seen that even though the cut-off frequency of all the three configurations

is now the same, the magnitude of the frequencies is differentfor different configurations

Applying the same low-pass frequency filter, previously used to obtain results pre-sented in Figure 5, to the time domain results shown in Figure 3, filtered time domainsignals were obtained for all the mesh configurations as seen in Figure 6. It can beobserved from Figure 6 that the filtered time domain signals are smoother in compari-son to their respective unfiltered counterparts (Fig. 3). It can also be observed fromFigure 6 that although all the filtered time domain signals have the same frequencies –proven by the filtered FFTs in Figure 5, the difference in smoothness among them,as observed in Figure 3, still persists such that the filtered signals from M2 are thesmoothest. Again, despite the signals containing the same band of frequencies, somecomponents present in M0 and M1 are not seen in M3. The latter observation is in con-trast to the expected model convergence behavior.

a) b)


Fig. 6. Filtered time domain signals of all the three mesh configurations: a) filtered timedomain signals of M0 mesh configuration (coarse mesh); b) filtered time domain signals of M1

mesh configuration (fine mesh); c) filtered time domain signals of M2 mesh configuration(finest mesh). It can be observed that although all the filtered time domain signals have

the same frequencies – proven by the filtered FFTs in Figure 5, the difference in smoothnessamong them, as observed in Figure 3, still persists

4. Discussion

As noted in Section 3, and clearly observed in Figure 4, the cut-off frequency in-creases as the mesh resolution becomes finer. This increase in the cut-off frequencyhappens because the numerical Brillouin zone increases with finer mesh resolution.Here, the numerical Brillouin zone is related to the structure of the model – and notthe material – such that the numerical Brillouin zone limitation is the short wavelengthlimitation that causes higher frequencies to be supported only by finer mesh resolu-tions. Thus, the numerical model acts like a filter, whose limit is determined by its

a) b)

c)


mesh resolution, that lets frequencies below the numerical Brillouin zone limit passthrough the model, and blocks all the frequencies that are higher than the numericalBrillouin zone limit. One of the key findings of the presented study indicates that thefiltering property of the model, which is a function of its mesh resolution, deformsthe passed frequencies such that some of them are amplified. The selective amplifica-tion property can be clearly observed in the filtered FFT results shown in Figure 5,wherein the limiting frequency is the same for all three configurations but the magni-tudes of frequencies are different. The filtering characteristics of the numerical modelcause the frequencies near the respective numerical Brillouin zones to be amplifiedbecause of amplitude-singularity. Therefore, magnitudes of the frequencies near thefilter limit, which is the cut-off frequency of M0, in Figure 5 are significantly higherfor M0 than the other two configurations. The amplified magnitudes of frequenciesnear the filtering limit observed in Figure 5 cause the filtered time domain signalfor M0 to be coarsest while the smoothest filtered time domain signals of M2 repre-sent presence of no amplified frequencies (as they have been filtered out by thelow-pass filter).

The amplified frequencies represent numerical errors. They can mask the AEsource characteristics, and it becomes vital to avoid or filter-out these in order to re-duce erroneous results. Thus, choosing the mesh resolution is important for AE sourcemodels; and although a coarse mesh is computationally cheaper, the results are rela-tively erroneous. A multi-scale approach becomes an attractive option in this predica-ment such that a balance can be achieved between computational efficiency and thequality of results.

5. Conclusions

In this work, we investigated the filtering impact of spatial discretization onbroadband AE source modeling. The investigation was based on the analysis of threedifferent mesh configurations using a commercial FEM solver.

Results for the three mesh configurations were presented. The data were pro-cessed using Fast Fourier Transform (FFT) and filtered to be subsequently analyzedin terms of the filtering effects of spatial discretization on AE source modeling.It was found that grids effectively work as low-pass filters with the cut-off frequenciesbeing a function of their spatial discretizations. The effective cut-off frequency – corre-sponding to the numerical Brillouin zone – induces the short wavelength limitation,hence frequencies near the numerical Brillouin zone are amplified in magnitude,followed by their blocking beyond that frequency. The amplified frequency waves


represent numerical errors and can mask the AE source characteristics, affecting qual-ity of the results.

The study reported here shows the significance of the choice of mesh resolution(size) for broadband wave phenomena. The latter are characteristic to seismic waves,acoustic emissions in soils, metals and composites during drilling, milling, miningand other processes. The mesh resolution, in seismic wave models, should be adjustedaccording to not just the topography but also the expected frequency range of thewaves and include the possible amplification and distortion of waves near the numeri-cal Brillouin zone limits. In particular, the cut-off property of the numerical Brillouinzone limit – leading to filtering out wave components beyond that frequency – causesalso distortions of propagating wave components below the limiting frequency. There-fore, additional margin should be considered when designing the model in order toavoid erroneous interpretations of the acquired signals. Specifically, a coarse meshchoice – although computationally cheaper – may neglect certain geometrical featuresof the terrain and also affect/amplify the frequencies of interest, leading to interpreta-tion errors resulting in poor predictions. A multi-scale approach thus becomes an at-tractive option for seismic wave simulation wherein a balance can be achieved betweencomputational efficiency and the quality of results.

This project has received funding from the European Union’s Horizon 2020 researchand innovation programme under the Marie Skłodowska-Curie grant agreementno. 764547. The second author acknowledges support from the National Science Centrethrough the OPUS project no. 2018/31/B/ST8/00753.

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grid dispersion and filtering in broadband acoustic

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